One shared property of G-CNNs and PDE-G-CNNs is that the input data usually needs to be lifted to a higher dimensional space. Take, for example, the case of image segmentation with a convolution neural network where we model/idealize the images as real-valued function on \(\mathbb {R}^2\) . If we keep the data as functions on \(\mathbb {R}^2\)  and want the convolutions within the network to be equivariant, then the only possible ones that are allowed are with isotropic kernels, [1]}. This type of short-coming generalizes to other symmetry groups as well [2]}. One can imagine that this is a constraint too restrictive to work with, and that is why we lift the image data.
